Properties

Label 22.2.c.a
Level $22$
Weight $2$
Character orbit 22.c
Analytic conductor $0.176$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 22.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.175670884447\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{10} q^{2} + ( - \zeta_{10}^{3} + \zeta_{10} - 1) q^{3} + \zeta_{10}^{2} q^{4} + (2 \zeta_{10}^{3} - 2) q^{5} + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 1) q^{6} - 2 \zeta_{10}^{2} q^{7} - \zeta_{10}^{3} q^{8} + (3 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{10} q^{2} + ( - \zeta_{10}^{3} + \zeta_{10} - 1) q^{3} + \zeta_{10}^{2} q^{4} + (2 \zeta_{10}^{3} - 2) q^{5} + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 1) q^{6} - 2 \zeta_{10}^{2} q^{7} - \zeta_{10}^{3} q^{8} + (3 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{9} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{10} + ( - 3 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4 \zeta_{10} + 2) q^{11} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 1) q^{12} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{13} + 2 \zeta_{10}^{3} q^{14} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{15} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{16} + ( - \zeta_{10}^{2} + \zeta_{10}) q^{17} + ( - 3 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 3 \zeta_{10}) q^{18} + (4 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{19} + ( - 2 \zeta_{10}^{2} - 2) q^{20} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2) q^{21} + (\zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10} - 3) q^{22} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2}) q^{23} + (\zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{24} + ( - 3 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{25} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{26} + ( - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 1) q^{27} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{28} + (4 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 4 \zeta_{10}) q^{29} + ( - 2 \zeta_{10} + 2) q^{30} - 2 \zeta_{10} q^{31} + q^{32} + (4 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 4 \zeta_{10} - 1) q^{33} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{34} + (4 \zeta_{10}^{2} + 4) q^{35} + (\zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{36} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10}) q^{37} + ( - 4 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 4) q^{38} + ( - 2 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 2) q^{39} + (2 \zeta_{10}^{3} + 2 \zeta_{10}) q^{40} + ( - 5 \zeta_{10}^{3} - \zeta_{10} + 1) q^{41} + ( - 2 \zeta_{10}^{2} + 4 \zeta_{10} - 2) q^{42} + (9 \zeta_{10}^{3} - 9 \zeta_{10}^{2} - 3) q^{43} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10} + 1) q^{44} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 8) q^{45} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{46} + (4 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{47} + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10}) q^{48} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{49} + (3 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 3) q^{50} + ( - 2 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 2 \zeta_{10}) q^{51} + ( - 2 \zeta_{10} + 2) q^{52} + ( - 4 \zeta_{10}^{2} + 8 \zeta_{10} - 4) q^{53} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 1) q^{54} + (2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 6 \zeta_{10}) q^{55} - 2 q^{56} + (2 \zeta_{10}^{2} - \zeta_{10} + 2) q^{57} + ( - 2 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{58} + ( - \zeta_{10}^{3} - 7 \zeta_{10}^{2} - \zeta_{10}) q^{59} + (2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{60} + ( - 4 \zeta_{10}^{2} + 4 \zeta_{10}) q^{61} + 2 \zeta_{10}^{2} q^{62} + ( - 2 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{63} - \zeta_{10} q^{64} + 4 q^{65} + (3 \zeta_{10}^{3} - 3 \zeta_{10} + 4) q^{66} + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} + 8) q^{67} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{68} + (2 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{69} + ( - 4 \zeta_{10}^{3} - 4 \zeta_{10}) q^{70} + ( - 4 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 4) q^{71} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} + 1) q^{72} + ( - \zeta_{10}^{3} + 12 \zeta_{10}^{2} - \zeta_{10}) q^{73} + (6 \zeta_{10} - 6) q^{74} + ( - 5 \zeta_{10}^{2} + 6 \zeta_{10} - 5) q^{75} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 4) q^{76} + (4 \zeta_{10}^{3} - 4 \zeta_{10} - 2) q^{77} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 2) q^{78} + ( - 12 \zeta_{10}^{2} + 6 \zeta_{10} - 12) q^{79} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{80} + (6 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 6 \zeta_{10}) q^{81} + (5 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 5) q^{82} + (5 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 5) q^{83} + (2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 2 \zeta_{10}) q^{84} + 2 \zeta_{10}^{3} q^{85} + (9 \zeta_{10}^{2} - 6 \zeta_{10} + 9) q^{86} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 2) q^{87} + (2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + \zeta_{10} - 2) q^{88} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} - 5) q^{89} + (2 \zeta_{10}^{2} + 6 \zeta_{10} + 2) q^{90} + (4 \zeta_{10} - 4) q^{91} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{92} + (2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 2) q^{93} + ( - 4 \zeta_{10}^{3} + 4) q^{94} + ( - 8 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 8 \zeta_{10}) q^{95} + ( - \zeta_{10}^{3} + \zeta_{10} - 1) q^{96} + (3 \zeta_{10}^{2} - 12 \zeta_{10} + 3) q^{97} - 3 q^{98} + ( - 13 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 4 \zeta_{10} + 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 4 q^{3} - q^{4} - 6 q^{5} + q^{6} + 2 q^{7} - q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 4 q^{3} - q^{4} - 6 q^{5} + q^{6} + 2 q^{7} - q^{8} + 7 q^{9} + 4 q^{10} - q^{11} + 6 q^{12} - 4 q^{13} + 2 q^{14} + 6 q^{15} - q^{16} + 2 q^{17} - 8 q^{18} - 5 q^{19} - 6 q^{20} - 12 q^{21} - 11 q^{22} - 4 q^{23} + q^{24} + 9 q^{25} + 6 q^{26} + 5 q^{27} + 2 q^{28} + 10 q^{29} + 6 q^{30} - 2 q^{31} + 4 q^{32} + 11 q^{33} + 2 q^{34} + 12 q^{35} - 8 q^{36} - 18 q^{37} + 10 q^{38} - 6 q^{39} + 4 q^{40} - 2 q^{41} - 2 q^{42} + 6 q^{43} + 4 q^{44} - 28 q^{45} - 4 q^{46} - 8 q^{47} - 4 q^{48} + 3 q^{49} - 11 q^{50} - 7 q^{51} + 6 q^{52} - 4 q^{53} + 4 q^{55} - 8 q^{56} + 5 q^{57} + 10 q^{58} + 5 q^{59} - 4 q^{60} + 8 q^{61} - 2 q^{62} + 16 q^{63} - q^{64} + 16 q^{65} + 16 q^{66} + 22 q^{67} + 2 q^{68} + 14 q^{69} - 8 q^{70} + 8 q^{71} + 7 q^{72} - 14 q^{73} - 18 q^{74} - 9 q^{75} - 10 q^{76} - 8 q^{77} - 16 q^{78} - 30 q^{79} + 4 q^{80} + 14 q^{81} - 7 q^{82} - 19 q^{83} + 8 q^{84} + 2 q^{85} + 21 q^{86} - 20 q^{87} - q^{88} - 10 q^{89} + 12 q^{90} - 12 q^{91} + 6 q^{92} + 2 q^{93} + 12 q^{94} - 10 q^{95} - 4 q^{96} - 3 q^{97} - 12 q^{98} - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\) \(13\)
\(\chi(n)\) \(\zeta_{10}^{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.809017 + 0.587785i 0.118034 + 0.363271i 0.309017 0.951057i −2.61803 1.90211i −0.309017 0.224514i −0.618034 + 1.90211i 0.309017 + 0.951057i 2.30902 1.67760i 3.23607
5.1 0.309017 + 0.951057i −2.11803 1.53884i −0.809017 + 0.587785i −0.381966 + 1.17557i 0.809017 2.48990i 1.61803 1.17557i −0.809017 0.587785i 1.19098 + 3.66547i −1.23607
9.1 0.309017 0.951057i −2.11803 + 1.53884i −0.809017 0.587785i −0.381966 1.17557i 0.809017 + 2.48990i 1.61803 + 1.17557i −0.809017 + 0.587785i 1.19098 3.66547i −1.23607
15.1 −0.809017 0.587785i 0.118034 0.363271i 0.309017 + 0.951057i −2.61803 + 1.90211i −0.309017 + 0.224514i −0.618034 1.90211i 0.309017 0.951057i 2.30902 + 1.67760i 3.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.2.c.a 4
3.b odd 2 1 198.2.f.e 4
4.b odd 2 1 176.2.m.c 4
5.b even 2 1 550.2.h.h 4
5.c odd 4 2 550.2.ba.c 8
8.b even 2 1 704.2.m.h 4
8.d odd 2 1 704.2.m.a 4
11.b odd 2 1 242.2.c.c 4
11.c even 5 1 inner 22.2.c.a 4
11.c even 5 1 242.2.a.f 2
11.c even 5 2 242.2.c.a 4
11.d odd 10 1 242.2.a.d 2
11.d odd 10 1 242.2.c.c 4
11.d odd 10 2 242.2.c.d 4
33.f even 10 1 2178.2.a.x 2
33.h odd 10 1 198.2.f.e 4
33.h odd 10 1 2178.2.a.p 2
44.g even 10 1 1936.2.a.n 2
44.h odd 10 1 176.2.m.c 4
44.h odd 10 1 1936.2.a.o 2
55.h odd 10 1 6050.2.a.ci 2
55.j even 10 1 550.2.h.h 4
55.j even 10 1 6050.2.a.bs 2
55.k odd 20 2 550.2.ba.c 8
88.k even 10 1 7744.2.a.cy 2
88.l odd 10 1 704.2.m.a 4
88.l odd 10 1 7744.2.a.cz 2
88.o even 10 1 704.2.m.h 4
88.o even 10 1 7744.2.a.bm 2
88.p odd 10 1 7744.2.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.2.c.a 4 1.a even 1 1 trivial
22.2.c.a 4 11.c even 5 1 inner
176.2.m.c 4 4.b odd 2 1
176.2.m.c 4 44.h odd 10 1
198.2.f.e 4 3.b odd 2 1
198.2.f.e 4 33.h odd 10 1
242.2.a.d 2 11.d odd 10 1
242.2.a.f 2 11.c even 5 1
242.2.c.a 4 11.c even 5 2
242.2.c.c 4 11.b odd 2 1
242.2.c.c 4 11.d odd 10 1
242.2.c.d 4 11.d odd 10 2
550.2.h.h 4 5.b even 2 1
550.2.h.h 4 55.j even 10 1
550.2.ba.c 8 5.c odd 4 2
550.2.ba.c 8 55.k odd 20 2
704.2.m.a 4 8.d odd 2 1
704.2.m.a 4 88.l odd 10 1
704.2.m.h 4 8.b even 2 1
704.2.m.h 4 88.o even 10 1
1936.2.a.n 2 44.g even 10 1
1936.2.a.o 2 44.h odd 10 1
2178.2.a.p 2 33.h odd 10 1
2178.2.a.x 2 33.f even 10 1
6050.2.a.bs 2 55.j even 10 1
6050.2.a.ci 2 55.h odd 10 1
7744.2.a.bm 2 88.o even 10 1
7744.2.a.bn 2 88.p odd 10 1
7744.2.a.cy 2 88.k even 10 1
7744.2.a.cz 2 88.l odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(22, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + 6 T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16 \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + T^{3} + 21 T^{2} + 11 T + 121 \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 5 T^{3} + 40 T^{2} + 50 T + 25 \) Copy content Toggle raw display
$23$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 10 T^{3} + 60 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$37$ \( T^{4} + 18 T^{3} + 144 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} + 24 T^{2} + 133 T + 361 \) Copy content Toggle raw display
$43$ \( (T^{2} - 3 T - 99)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} + 64 T^{2} + 192 T + 256 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} + 96 T^{2} - 256 T + 256 \) Copy content Toggle raw display
$59$ \( T^{4} - 5 T^{3} + 60 T^{2} + \cdots + 3025 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} + 64 T^{2} - 192 T + 256 \) Copy content Toggle raw display
$67$ \( (T^{2} - 11 T - 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 8 T^{3} + 24 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$73$ \( T^{4} + 14 T^{3} + 136 T^{2} + \cdots + 17161 \) Copy content Toggle raw display
$79$ \( T^{4} + 30 T^{3} + 540 T^{2} + \cdots + 32400 \) Copy content Toggle raw display
$83$ \( T^{4} + 19 T^{3} + 186 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$89$ \( (T^{2} + 5 T - 25)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 3 T^{3} + 144 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
show more
show less